On the top eigenvalue of heavy-tailed random matrices

We study the statistics of the largest eigenvalue lambdaₘax of N x N random matrices with unit variance, but power-law distributed entries, P (M₈₉) ~ |M₈₉|^-1-mu. When mu > 4, lambdaₘax converges to 2 with Tracy-Widom fluctuations of order N^-2/3. When mu < 4, lambdaₘax is of order N^2/mu-1/2 and is governed by Fr\'echet statistics. The marginal case mu=4 provides a new class of limiting distribution that we compute explicitely. We extend these results to sample covariance matrices, and show that extreme events may cause the largest eigenvalue to significantly exceed the Marcenko-Pastur edge. Connections with Directed Polymers are briefly discussed.